Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 21-28, find the indicated derivative. $$ y^{\prime} \text { where } y=(x+\sin x)^{2} $$

Step-by-Step Solution

Answer

The derivative is \( y' = 2(x + \sin x)(1 + \cos x) \).

Step 1: Identify the differentiation rule to use

The given function is \[ y = (x + \sin x)^2. \]Since we have a composition of functions (a square function and another function within it), we will use the chain rule in differentiation.

Step 2: Apply the chain rule

According to the chain rule, if we have a composite function \( y = (g(x))^2 \), then the derivative \( y' \) is given by: \[ y' = 2g(x) \cdot g'(x). \]Here, \( g(x) = x + \sin x \).

Step 3: Find the derivative of the inner function

Calculate the derivative of \( g(x) = x + \sin x \), which is:\[ g'(x) = \frac{d}{dx}(x) + \frac{d}{dx}(\sin x) = 1 + \cos x. \]

Step 4: Combine results to find the derivative of the original function

Using the chain rule, substitute \( g(x) \) and \( g'(x) \) back into the expression:\[ y' = 2(x + \sin x)(1 + \cos x). \]

Step 5: Simplify the expression, if necessary

The derivative can be expressed as:\[ y' = 2(x + \sin x)(1 + \cos x), \]which is already simplified, and that is our final answer.

Key Concepts

Chain Rule

The chain rule is a fundamental tool in calculus used to differentiate composite functions. A composite function is when one function is nested inside another function. To differentiate such a function, the chain rule comes into play, allowing us to break down the process into manageable steps. In essence, the chain rule formula is: If you have a function of the form \( y = (g(x))^n \), the derivative \( y' \) is given by \[ y' = n(g(x))^{n-1} \cdot g'(x). \] Here, \( g(x) \) represents the inner function, and its derivative, \( g'(x) \), is crucial in determining the derivative of the whole function. The chain rule helps us manage complexity by allowing us to focus on one layer of the function at a time. This is especially useful when dealing with functions involving trigonometric expressions, exponential functions, or any situation where multiple operations are layered.

Differentiation

Differentiation is the process used in calculus to find the derivative of a function. A derivative represents the rate at which a function is changing at any given point, essentially providing the slope of the tangent line to the function at that point. There are basic rules of differentiation, such as:

• The power rule: \( \frac{d}{dx}(x^n) = nx^{n-1} \)
• The sum rule: \( \frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x) \)
• And importantly for our exercise, the chain rule, which handles composite functions.

Applying these rules allows us to deconstruct complex expressions into simpler parts, enabling accurate computation of their derivatives.

Composite Functions

Composite functions are created when one function is evaluated inside another function. For example, in the function \( y = (x + \sin x)^2 \), the expression \( x + \sin x \) is the inner function, and raising to the power of 2 is the outer function. When working with composite functions, it's essential to identify the inner and outer parts clearly to apply the appropriate rules for differentiation. The chain rule is particularly vital here because it addresses these multi-layered functions effectively.The key steps include:

• Identifying the inner function (\( g(x) = x + \sin x \)) and the outer function (raising it to the power 2).
• Differentiating the inner function first (finding \( g'(x) \)).
• Then using the chain rule to combine these derivatives correctly and get the complete derivative of the composite function.

This structured approach is what gives the power to tackle more complex calculus problems, simplifying the solution within the layers.